[Insight-users] Implementation Finite Element Method (FEM) in ITK

[Cristina Oyarzun]

Hello Brian,

thank you for your help! Now my doubts are solved.

Greetings,

Cristina

On Wed, Feb 2, 2011 at 2:17 AM, brian avants <stnava at gmail.com> wrote:

> hi jeremy , cristina
>
> here is a brief latex sketch of the method used in the
> FEMCrankNicolsonSolver --
>
> \text{matrix form parabolic PDE, heat equation type} \\
> M\frac{ \partial U}{\partial t} +  K U  = f \\
> \frac{ M U_t - M  U_{t-1}}{\delta_t} +  K U  = f \\
> \frac{ M U_t - M  U_{t-1} +  \delta_t K U }{\delta_t} = f \\
> \text{ here i make the decision to represent $U$ and $f$ at $t$ rather
> than $t-1$ i believe this is backward Euler}\\
> ( M  + \delta_t K ) U_t  =  M  U_{t-1} + \delta_t f_t \\
> \\
> \text{other choices are possible, e.g. } \\
> \text{ Crank-Nicolson discretization is based on averaging forward and
> backward Euler } \\
> ( M + \alpha \delta_t K ) U_t = (  M - ( 1- \alpha ) \delta_t  K )
> U_{t-1} + f \\
>  f  =  \delta_t ( \alpha f_t + ( 1 - \alpha ) f_{t-1} ) \\
> \\
> \text{if}~ \alpha = 1 \text{ backward Euler }\\
> ( M +  \delta_t K ) U_t = M  U_{t-1} + \delta_t f \\
> \\
> \text{if}~ \alpha = 0 \text{ forward Euler }\\
>  M U_t =(  M - \delta_t  K ) U_{t-1} + \delta_t f_{t-1}
>
> hopefully this is helpful.
>
> brian
>
>
>
>
> On Tue, Feb 1, 2011 at 4:01 PM, Jeremy Bournesel
> <jeremy.bournesel at gmail.com> wrote:
> > Hi,
> > I have a decent background regarding FEM in the engineering domain and
> have
> > started looking at it for registering medical images as well.
> > I ran one of the ITK deformable registration demos, read the Software
> Guide
> > and the ITK Powerpoint Presentations, however there is still some magic
> > going on that I don't completely get.
> > From what I understand ITK is using a uniform grid/mesh (if a custom one
> > wasn't supplied) and then calculates the deformation for every element at
> > the nodal points (iteratively).
> > I couldn't find any information on how the exact process is working
> (besides
> > some high level slides of the type K U = F).
> > It'd be awesome if someone can point me to the right document or explain
> it
> > to me.
> > Thanks,
> > Jeremy
> >
> >
> >
> >
> >
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